C₇×#10523 #13187739

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10523

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

382 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,13,17)(10,18,15,11)(12,19,22,16)(14,21,23,20)(24,25)(26,34)(27,32)(28,31)(29,33)(30,36)(35,39)(37,38);(8,10,13,15)(9,14,17,23)(11,19,18,16)(12,21,22,20)(24,26)(25,30)(27,36)(28,37)(29,35)(31,39)(32,34)(33,38);(8,11,12,14,13,18,22,23)(9,15,16,21,17,10,19,20)(25,31)(26,35)(28,29)(32,33)(36,37)(38,39)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3511);